The fifth order polynomial f(x) = 3x^5 + 5x^4 + x³/3+1 has one real root, x*

-1.644666506.

(a) Plot the polynomial on the interval [-2, 1]. Hand in your plot with your

assignment.

(b) Use Newton's method with a tolerance of 10-¹0 to find the root. By trying

several different initial values, x₁, determine in what interval must you choose

₁ in order achieve convergence to x*. Explain why this is the case.

(c) Determine a function g(x) so that the fixed point iteration method is guaran-

teed to converge to r*. Explain why convergence is guaranteed for your choice

of g.

(d) Using the fixed_point (...) function provided, apply the fixed point method

with this g(x) and a tolerance of 10-10.

(e) Choose ₁ -2 and compare the speed of convergence (number of iterations

to achieve tolerance) of the two methods. Does this agree with the content of

the convergence theorems for the two methods?